1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

Ultrafilters and Ultraprod ucts Appendix A Since we need them throughout this book, here's a brief account of ultrafil- ters and ...

446 A. Ultrafilters and Ultraproducts Example A.3. Let I be an infinite set. The Frechet filter on I is the family of all subset ...

A. UltraE.lters and Ultraproducts 447 by rp(f) = ~im f(i). i->U It is not too hard to check that rp is a character (i.e., a - ...

448 A. Ultrafi.lters and Ultraproducts Lemma A.9. The ultraproduct Mw is a von Neumann algebra and the faithful tracial state rw ...

Appendix B Operator Spaces, Completely Bounded Maps and Duality Here is a crash course on the operator space theory needed in th ...

450 B. Operator Spaces ::::; ll'Pll sup{ll L~i1JjXi,jll: L l~il^2 =1=L117jl 2 } i,j i j ::::; ll'Pllll[xi,j]ll· If X c A is an o ...

B. Operator Spaces 451 For the proof, we need the following lemma. Lemma B.6. For any operators a, b;:::: 0 and x, we have [ :* ...

452 B. Operator Spaces Proof. Without loss of generality, we may assume that A is unital. Since cp: X ----+ IIB(7t) is c.c., it ...

B. Operator Spaces 453 Corollary B.9. Let E C A be an operator system and <p: E ~ JEB(7i) be a unital self-adjoint map (i.e., ...

454 B. Operator Spaces Proof. Let B c IIB(H). Then, Corollary B.9 provides a u.c.p. map 1/J': E----* IIB(H) with ll'P -1/J'llcb ...

B. Operator Spaces 455 where 1-i = EBmEN EBxEBallm(X) £~. Unless otherwise stated, we always as- sume that the dual space X* is ...

456 B. Operator Spaces Remark B.14 (Technical observation). Let Ebe a finite-dimensional op- erator space and A be a C -algebra. ...

B. Operator Spaces 457 there exists a net {x.>.h of norm-one elements in Mn(X) which converges toxin the cr(Mn(X),Mn(X))-topo ...

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Appendix C Lifting Theorems Throughout this appendix, we assume that J is a closed two-sided ideal in a unital C* -algebra B and ...

460 C. Lifting Theorems To prove the claim, we proceed by induction. First, set 1/;1 = 1/Ji. Sup- pose now that we have construc ...

C. Lifting Theorems 461 Theorem C.4. Let J be a closed two-sided ideal in a unital C* -algebra B and 7r: B ---+ B / J be the quo ...

462 C. Lifting Theorems Remark C.5. Though one can almost always get around it by working locally, a convenient generalization i ...

Positive Definite Functions, Cocycles and Schoenberg's Theorem Appendix D Unitary representations. As in Section 2.5, r denotes ...

464 D. Positive Definite Functions unitary representation 7r of r on H such that cp(s) = P'h'.7r(s)l'h'. for every s E I'. There ...